### A NOTE ON RELATION BETWEEN FUZZY INTERIOR AND FUZZY CLOSURE WITH EXTENDED DEFINITION OF FUZZY SET

Research Scholar, Department of Mathematical Sciences
Bodoland University, Kokrajhar, Assam, India

Abstract
The main purpose of this article is to construct the relation between fuzzy interior and fuzzy closure with extended definition of fuzzy set. We have discussed on fuzzy interior operator and fuzzy closure operator with extended definition of fuzzy set.

Key words: Fuzzy membership function, Fuzzy reference function, Fuzzy interior, Fuzzy closure

1.   Introduction
Fuzzy set theory was discovered by Zadeh (1965). Chang (1968) introduced fuzzy topology. After the introduction of fuzzy sets and fuzzy topology, several researches were conducted on the generalizations of the notions of fuzzy sets and fuzzy topology.  The theory of fuzzy sets actually has been a generalization of the classical theory of sets in the sense that the theory of sets should have been a special case of the theory of fuzzy sets. But unfortunately it has been accepted that for fuzzy set A and its complement AC, neither A AC is empty set nor A AC is the universal set. Whereas the operations of union and intersection of crisp sets are indeed special cases of the corresponding operation of two fuzzy sets, they end up giving peculiar results while defining A AC and A AC. In this regard Baruah (1999, 2011, 2011) has forwarded an extended definition of fuzzy sets which enable us to define complement of fuzzy sets in a way that give us A AC is empty set and A Ais universal set.

In this article we would discussed relation between fuzzy interior and fuzzy closure with extended definition of fuzzy set by giving counter example. Also we would discuss on fuzzy interior operator and fuzzy closure operator with extended definition of fuzzy set.

2.   Objective
It seems that the existing definition of complement of fuzzy sets were not propose within the mathematical frameworks, by observing this, through this paper, an attempt is made to construct the relation between fuzzy interior and fuzzy closure with extended definition of fuzzy set.

3.   Methodology
In this article, extended definition of fuzzy set is used as key method to prove the theorems on fuzzy interior and fuzzy closure. Depending on data collected on complement of fuzzy set this work has been prepared.

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